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  title: 'Prime element of a ring',
  clickbait: 'Despite the name, "prime" in ring theory refers not to elements which are "multiplicatively irreducible" but to those such that if they divide a product then they divide some term of the product.',
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  text: '[summary:  A prime element of a [3gq ring] is one such that, if it divides a product, then it divides (at least) one of the terms of the product.]\n\n[summary(Technical): Let $(R, +, \\times)$ be a [3gq ring] which is an [5md integral domain]. We say $p \\in R$ is *prime* if, whenever $p \\mid ab$, it is the case that either $p \\mid a$ or $p \\mid b$ (or both).]\n\nAn element of an [-5md] is *prime* if it has the property that $p \\mid ab$ implies $p \\mid a$ or $p \\mid b$.\nEquivalently, if its generated [ideal_ring_theory ideal] is [prime_ideal prime] in the sense that $ab \\in \\langle p \\rangle$ implies either $a$ or $b$ is in $\\langle p \\rangle$.\n\nBe aware that "prime" in ring theory does not correspond exactly to "[4mf prime]" in number theory (the correct abstraction of which is [5m1 irreducibility]). \nIt is the case that they are the same concept in the ring $\\mathbb{Z}$ of [48l integers] ([5mf proof]), but this is a nontrivial property that turns out to be equivalent to the [-5rh] ([alternative_condition_for_ufd proof]).\n\n# Examples\n\n# Properties\n\n- Primes are always [5m1 irreducible]; a proof of this fact appears on the [5m1 page on irreducibility], along with counterexamples to the converse.\n- ',
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